One of the most promising New Plant Breeding Techniques is genome editing (also called gene editing) with the help of a programmable site-directed nuclease (SDN). In this review, we focus on SDN-1, which is the generation of small deletions or insertions (indels) at a precisely defined location in the genome with zinc finger nucleases (ZFN), TALENs, or CRISPR-Cas9. The programmable nuclease is used to induce a double-strand break in the DNA, while the repair is left to the plant cell itself, and mistakes are introduced, while the cell is repairing the double-strand break using the relatively error-prone NHEJ pathway. From a biological point of view, it could be considered as a form of targeted mutagenesis. We first discuss improvements and new technical variants for SDN-1, in particular employing CRISPR-Cas, and subsequently explore the effectiveness of targeted deletions that eliminate the function of a gene, as an approach to generate novel traits useful for improving agricultural sustainability, including disease resistances. We compare them with examples of deletions that resulted in novel functionality as known from crop domestication and classical mutation breeding (both using radiation and chemical mutagens). Finally, we touch upon regulatory and access and benefit sharing issues regarding the plants produced.